INTERNATIONAL JOURNAL OF c© 2012 Institute for ScientificNUMERICAL ANALYSIS AND MODELING, SERIES B Computing and InformationVolume 3, Number 3, Pages 207–223

COALESCENCE CUBIC SPLINE FRACTAL

INTERPOLATION SURFACES

ARYA KUMAR BEDABRATA CHAND

Abstract. Fractal geometry provides a new insight to the approximation and modelling ofscientific data.This paper presents the construction of coalescence cubic spline fractal interpolationsurfaces over a rectangular grid D through the corresponding univariate basis of coalescence cubicfractal splines of Type-I or Type-II. Coalescence cubic spline fractal surfaces are self-affine or non-self-affine in nature depending on the iterated function systems parameters of these univariatefractal splines. Upper bounds of L∞-norm of the errors between between a coalescence cubicspline fractal surface and an original function f ∈ C4[D], and their derivatives are deduced.Finally, the effects of free variables, constrained free variables and hidden variables are discussedfor coalescence cubic spline fractal interpolation surfaces through suitably chosen examples.

Key words. Fractals, Iterated Function System, Fractal Interpolation Surface, Cardinal CubicSpline, Hidden Variables, CHFIS, Non-self-affine and Surface Approximation

1. Introduction

The theory of fractal interpolation has become a powerful tool in applied sci-ence and engineering since Barnsley [1] introduced fractal interpolation function(FIF) using the theory of iterated function system (IFS). The attractor of an IFSis the graph of a FIF that interpolates a given set of data points. Fractal interpola-tion constitutes an advance in techniques of approximation in the sense that thesefunctions used are not necessarily differentiable, and show the rough aspect of real-world signals [2–4]. For smooth curve approximation through fractal methodology,Barnsley and Harrington [5] initiated the construction of a differentiable FIF orCr-FIF f that interpolates the prescribed data if values of f (k), k = 1, 2, . . . , r, areassigned at the initial end point of the interval. Fractal splines with general bound-ary conditions are studied recently [6, 7]. The power of fractal methodology allowsus to generalize almost any other interpolation techniques, see for instance [8, 9].

Fractal surfaces are proved to be useful to approximate various type of surfaces inmaterial science, ocean engineering, geology, chemistry, physics, image processingand computer graphics. Massopust [10] was first to put forward the construction offractal interpolation surfaces (FISs) wherein he assumed the surface as triangularsimplex and interpolation points on the boundary to be co-planar. In view of lackof flexibility in this construction, Geronimo and Hardin [11] and Zhao [12] have gen-eralized the construction of FIS by allowing more general boundary data. Xie andSun [13] constructed bivariate FIS on rectangular grids with arbitrary contractionfactors and without any condition on boundary points. Dalla [14] extemporisedthis construction by using collinear boundary points and demonstrated that the at-tractor is a continuous FIS. Further research and developments on FISs in variousdirections are discussed by Massopust [15], Bouboulis, et al. [16, 17], Chand andNavascues [18] , Metzler and Yunb [19]. However, all the constructions mentionedabove lead to self-affine fractal surfaces.

Received by the editors May 20, 2011 and, in revised form, March 25, 2012.2000 Mathematics Subject Classification. 28A80, 41A05, 41A15, 41A25, 65D10, 65D17.

207

208 A. K. B. CHAND

The term ‘hidden variable’ was introduced by Barnsley et al. [20] and Masso-pust [21]. A hidden variable FIF (HFIF) is more diverse, appealing and irregularthan a FIF for the same set of interpolation data as functional values of a HFIF con-tinuously depend on all the defining IFS parameters. Since a HFIF is the projectionof a vector valued function, it is usually non-self-affine in nature. Bouboulis andDalla [22] have constructed hidden variable vector valued FIFs on random gridsin R

2. Chand and Kapoor [23] have introduced the coalescence hidden variableFIF and studied their stability analysis [24]. A non-diagonal IFS that generatesboth self-affine and non-self-affine FIS simultaneously, depending on the free vari-ables and constrained variables on a general set of interpolation data is constructedin [25]. The attractor of such an IFS is called the coalescence hidden-variable frac-tal interpolation surface (CHFIS). A CHFIS is a preferred choice for the studyof highly uneven surfaces such as clouds, sea surfaces, surfaces of rocks, tsunamiwaves, etc. The quantification of smoothness of such surfaces in terms of Lipschitzexponent of its corresponding CHFIS is investigated recently in [26]. This paperaims to develop the theory of coalescence cubic spline FISs (CCFISs), to study theirconvergence results, and to validate the effects of IFS parameters on the shape ofa CCFIS.

In Section 2, we discuss basics of coalescence hidden variable FIFs (CHFIFs),construction of cubic spline CHFIFs and cardinal cubic CHFIFs. An estimate ofthe error bound of the cubic spline CHFIF with the original function is obtainedin this section. The construction of CCFISs is carried out in Section 3 throughtensor product of cardinal cubic spline CHFIFs of Type-I or Type-II. Upper boundsof L∞-norm of the errors between a cubic spline CHFIF and the original functionf ∈ C4[D], and their derivatives are deduced in Section 4. Finally, the effects of IFSparameters on a CCFIS are illustrated through various suitably chosen examples.

2. Cubic Spline CHFIFs

We discuss the basics of CHFIFs through IFS theory in Section 2.1. The con-struction of cubic spline CHFIFs and cardinal cubic spline CHFIFs of Type-I orType-II are described respectively in Section 2.2 and Section 2.3. Upper boundsof L∞-norm of the errors between a cubic spline CHFIF and an original function,and their derivatives are estimated in Section 2.4.

2.1. Basics of CHFIFs. Let ∆t : t0 < t1 < · · · < tN be a partition of aninterval I = [t0, tN ] ⊂ R and {(tj, xj) ∈ I × R : j = 0, 1, 2, . . . , N} be a set ofinterpolation data points. This data set is extended to a generalized set of data{(tj, xj , ξj) ∈ R

3 : j = 0, 1, 2, . . . , N} with real parameters ξj , j = 0, 1, 2, . . . , N .Let Lj : I −→ Ij = [tj−1, tj ] be a contraction map satisfying

(2.1) Lj(t0) = tj−1, Lj(tN ) = tj for j = 1, 2, . . . , N.

Let Fj : I × R2 −→ R

2 be a vector valued function satisfying

(2.2)

{

Fj(t0, x0, ξ0) = (xj−1, ξj−1), Fj(tN , xN , ξN ) = (xj , ξj),

d(Fj(t, x, ξ), Fj(t∗, x∗, ξ∗)) ≤ τjdE((x, ξ), (x

∗, ξ∗)),

for j = 1, 2, . . . , N, where (t, x, ξ), (t∗, x∗, ξ∗) ∈ I × R2, 0 ≤ τj < 1, d is the sup.

metric on I×R2, and dE is the Euclidean metric on R

2. In order to define a CHFIF,functions Lj and Fj are chosen such that Lj(t) = ajt+ bj and

(2.3) Fj(t, x, ξ) = Aj(x, ξ)T + (pj(t), qj(t))

T ≡ (F 1j (t, x, ξ), F

2j (t, ξ))

T ,

COALESCENCE CUBIC SPLINE FIS 209

where Aj is an upper triangular matrix

(

αj βj0 γj

)

and pj(t), qj(t) are continuous

functions having two free parameters. These parameters can be evaluated by using(2.2). We take αj as a free variable with |αj | < κ < 1 and βj as a constrained freevariable with respect to a free hidden variable γj (|γj | < 1) such that |βj |+ |γj | <κ∗ < 1. The generalized IFS for construction of a CHFIF corresponding to thedata {(tj, xj , ξj)| j = 0, 1, . . . , N} is now defined as

(2.4) {R3;ωj(t, x, ξ) = (Lj(t), Fj(t, x, ξ)), j = 1, 2, . . . , N}.

It is easy to see that the IFS (2.4) is hyperbolic with respect to a metric ρ that isequivalent to the Euclidean metric on R

3. Hence, there exists a unique nonempty

compact set G ⊆ R3, called as an attractor of the IFS (2.4) such that G =

N⋃

j=1

ωj(G).

This attractor G provides the existence of a unique vector valued interpolant f inthe following proposition.

Proposition 2.1. [23] The attractor G of the IFS defined by (2.4) is the graph ofthe continuous vector valued function f : I −→ R

2 such that f(tj) = (xj , ξj) for allj = 0, 1, 2, . . . , N, i.e., G = {(t, x, ξ) : t ∈ I and f(t) = (x(t), ξ(t))}.

Proposition 2.1 gives that the graph of the vector valued function f(t) = (f1(t), f2(t))is the attractor of the IFS (2.4) if and only if the fixed point f of Read-Bajraktarevicoperator T on F = {h∗ ∈ C(I,R2) | h∗(t0) = (x0, ξ0) and h

∗(tN ) = (xN , ξN )} sat-isfies

(2.5) Tf(t) = f(t) = Fj(L−1j (t), f(L−1

j (t))), t ∈ Ij , j = 1, 2, . . . , N.

The image Tf of the vector valued function f can be considered component wiseas (T1f1, T2f2), where T1 and T2 are component wise Read-Bajraktarevic operatorsfrom I to R. The CHFIF is the function f1(t) obtained as projection of the attractorG on R

2 as {(t, x) : (t, x, ξ) ∈ G} = Graph of f1(t). The CHFIF f1 satisfies(2.6)T1f1(Lj(t)) = f1(Lj(t)) = F 1

j (t, f1(t), f2(t)) = αjf1(t) + βjf2(t) + pj(t), t ∈ I.

Similarly, {(t, ξ) : (t, x, ξ) ∈ G} = Graph of f2(t), where f2(t) is a self-affine fractalfunction that interpolates the data {(tj , ξj) : j = 0, 1, . . . , N}, and f2 satisfies

(2.7) T2f2(Lj(t)) = f2(Lj(t)) = F 2j (t, f2(t)) = γjf2(t) + qj(t), t ∈ I.

CHFIFs are generally non-self-affine in nature as the projection of the attractoris not always union of affine transformations of itself. But if either ξj = xj(j =0, 1, . . . , N) and αj + βj = γj or βj = 0 for all j = 1, 2, . . . , N , then CHFIF f1obtained as the projection on R

2 of the attractor of the IFS (2.4) coincides with aself-affine fractal function f2 for the same interpolation data. Hence, the CHFIF isself-affine in this case.

2.2. Construction of cubic spline CHFIFs. A function S(t) is said to be acubic spline on a grid t0 < t1 < · · · < tN if it satisfies (i) S(t) is a polynomialof degree 3 on each subinterval [tj−1, tj ] (ii) S

(r)(t) is continuous on [t0, tN ] forr = 0, 1, 2. The cubic spline is called of Type-I (Type-II) if S′(t0) and S

′(tN ) (S′′(t0)and S′′(tN ) respectively) are prescribed with the interpolation data. The existenceof a Cr-differentiable or spline FIF is guaranteed by the following proposition:

Proposition 2.2. [5] Let {(tj , xj)|j = 0, 1, 2, . . . , N} be the interpolation datawith t0 < t1 < t2 < · · · < tN . Let Lj(t) = ajt + bj satisfies (2.1) and Fj(t, x) =

210 A. K. B. CHAND

αjx + qj(t) for j = 1, 2, . . . , N. Suppose for some integer r > 0, |αj | < arj , andqj ∈ Cr[t0, tN ]; j = 1, 2, . . . , N. Let for k = 1, 2, . . . , r:

Fj,k(t, x) =αjx+ q

(k)j (t)

akj, x0,k =

q(k)1 (t0)

ak1 − α1, xN,k =

q(k)N (tN )

akN − αN

.

If Fj−1,k(tN , xN,k) = Fj,k(t0, x0,k) for j = 2, 3, . . . , N and k = 1, 2, . . . , r, then

{(Lj(t), Fj(t, x))}Nj=1 determines a FIF f ∈ Cr[x0, xN ] and f (k) (k = 1, 2, . . . , r)

is the FIF determined by {(Lj(x), Fj,k(t, x))}Nj=1.

Based on Proposition 2.2, if the values f r(t0), r = 0, 1, 2, . . . r are known, thena Cr-FIF can be constructed through an algebraic method [5]. For a cubic splineC2-CHFIF with general boundary condition, we proceed as follows:

Let G∗ = {g∗ ∈ C2(I,R2) | g∗(t0) = (x0, ξ0) and g∗(tN ) = (xN , ξN )}. Let ρ∗ be

the C2-norm on G∗. Define the Read-Bajraktarevic operator T ∗ on vector valuedfunctions space (G∗, ρ∗) for t ∈ Ij , and j = 1, 2, . . . , N as

T ∗g∗(t) = Fj(L−1j (t), g∗(L−1

j (t))) ≡ A∗jg

∗(L−1j (t)) + a2j(pj(L

−1j (t)), qj(L

−1j (t)))T ,

where Lj(t) = ajt+ bj satisfies (2.1), g∗(t) = (g∗1(t), g∗2(t))

T , A∗j is an upper trian-

gular matrix a2j

(

αj βj0 γj

)

and pj , qj are suitably chosen cubic polynomials. Note

that ‖A∗j‖1 < a2j max{κ, κ∗} < a2j for j = 1, 2, . . . , N, which is a necessary condition

for differentiability of a fractal function according to Proposition 2.2. Therefore,T ∗ is a contractive operator on (G∗, ρ∗). Since (G∗, ρ∗) is complete, the fixed pointf = (f1, f2) of T

∗ satisfies the implicit relation:

(2.8) f(Lj(t)) = A∗jf(t) + a2j(pj(t), qj(t))

T , j = 1, 2, . . . , N.

In general, f ′ and f ′′ interpolate data different from the generalized data. In par-ticular, the components f ′′

1 is an affine CHFIF and f ′′2 is an affine fractal functions

as p′′j (t) and q′′j (t) are affine for j = 1, 2, . . . , N . Let qj(t) =3∑

l=0

q∗ljtl, pj(t) =

3∑

l=0

p∗ljtl, j = 1, 2, . . . , N, where the 8N coefficients q∗lj , p

∗lj have to be determined

suitably such that f interpolates the generalized interpolation data. The coefficients

of qj(t) are computed first in order to evaluate f(k)2 (t0) and f

(k)2 (tN ), k = 1, 2 for the

computation of pj(t) in the construction of the cubic spline CHFIF. The continuity

of f(k)2 on I gives f

(k)2 (Lj+1(t0)) = f

(k)2 (Lj(tN )) for k = 0, 1, 2, j = 1, 2, . . . , N − 1

Hence, we can write for k = 0, 1, 2, j = 1, 2, . . . , N − 1:

(2.9) akj+1{γj+1f(k)2 (t0) + q

(k)j+1(t0)} = akj {γjf

(k)2 (tN ) + q

(k)j (tN )}.

At the end points t0 and tN , first and second derivatives of f2 satisfy

f(k)2 (t0) = a2−k

1 {γ1f(k)2 (t0) + q

(k)1 (t0)}, k = 1, 2,

f(k)2 (tN ) = a2−k

N {γNf(k)2 (tN ) + q

(k)N (tN )}, k = 1, 2.

(2.10)

The interpolation conditions are

(2.11) f2(tj) = ξj , j = 0, 1, . . . , N.

From (2.9)-(2.11), the total number of the conditions for f2 to interpolate theprescribed data is 4N + 2, where as the total number of unknowns associatedwith f2 is 4N + 4. If two suitable relations involving values of fractal function f2and values of its derivatives at end points of [t0, tN ] are chosen, then f

(k)2 (t0) and

COALESCENCE CUBIC SPLINE FIS 211

f(k)2 (tN ) for k = 1, 2, and q∗lj (l = 0, 1, 2, 3; j = 1, 2, . . . , N) can be evaluated fromthese system of equations. These derivatives values will be used in the construction

of the cubic spline CHFIF f1. Analogously the values of f(k)1 (t0) and f

(k)1 (tN ) for

k = 1, 2, and coefficients of pj(t) can be determined from

(2.12)

akj+1{αj+1f(k)1 (t0) + βj+1f

(k)2 (t0) + p

(k)j+1(t0)} = akj {αjf

(k)1 (tN )

+ βjf(k)2 (tN ) + p

(k)j (tN )}, k = 0, 1, 2, j = 1, 2, . . . , N − 1,

f(k)1 (t0) = a2−k

1 {α1f(k)1 (t0) + β1f

(k)2 (t0) + p

(k)1 (t0)}, k = 1, 2,

f(k)1 (tN ) = a2−k

N {αNf(k)1 (tN ) + βNf

(k)2 (tN ) + p

(k)N (tN )}, k = 1, 2,

f1(tj) = xj , j = 0, 1, . . . , N,

if we assume two suitable relations involving the values of f1 and the values of itsderivatives at end points of [t0, tN ]. Hence, the required cubic spline CHFIF f1interpolating the prescribed data is constructed as projection of the attractor ofthe following IFS:(2.13)

{R3;ωj(t, x, ξ) = (Lj(t), Fj(t, x, ξ) ≡ (F 1j (t, x, ξ), F

2j (t, ξ))), j = 1, 2, . . . , N},

where F 1j (t, x, ξ) = a2j(αjx + βjξ + pj(t)), F

2j (t, ξ) = a2j (γjξ + qj(t)), |αj | < κ < 1,

|βj | + |γj | < κ∗ < 1, qj(t) and pj(t), j = 1, 2, . . . , N are cubic polynomials withcoefficients q∗lj and p

∗lj respectively, computed by solving linear independent systems

of equations. In this construction, an infinite number of spline CHFIFs of Type-Ior Type-II can be constructed depending upon the IFS parameters: αj , βj , γj , ξjand the boundary conditions of self-affine fractal function f2.

2.3. Cardinal cubic spline CHFIFs.

Definition 2.1. A function f1i(t) is called a cardinal cubic spline CHFIF if (i) f1iis a CHFIF associated with the generalized set of data points {(tj , δi,jξi,j) ∈ R

3 |i =0, 1, . . . , N} with mesh ∆t, that is to say

(2.14)f1i(tj) = δi,j = 1, i = j,

= 0, i 6= j,

}

∀ i, j = 0, 1, 2, . . . , N.

(ii) f1i ∈ C2[t0, tN ], (iii) the corresponding IFS {I × R2;wi,j(t, x, ξ) : j =

1, 2, . . . , N} generates the cardinal natural CHFIF f1i(t) with free variables αi,j,constrained free variables βi,j, and hidden variables: γi,j, ξi,j, the boundary condi-tions of self-affine fractal function f2i(t).

In the construction of the cardinal cubic CHFIF f1i, i = 0, 1, . . . , N , we havetaken αi,j = αj , βi,j = βj , γi,j = γj , j = 1, 2, . . . , N for simplicity in the IFS {I ×R

2 ; wi,j(t, x, ξ) = (Lj(t), Fi,j(t, x, ξ) ≡ (F 1i,j(t, x, ξ), F

2i,j(t, ξ))), j = 1, 2, . . . , N}

(cf. (2.13)). In order to construct a basis of cubic spline CHFIFs of Type-I, weneed f1

′i(t0) = f1

′i(tN ) = 0 for i = 0, 1, . . . , N, and two more fractal splines as f1−1

and f1N+1 such that

f1−1(tj) = 0, j = 0, 1, . . . , N ; f1′−1(t0) = 1, f1

′−1(tN ) = 0,

f1N+1(tj) = 0, j = 0, 1, . . . , N ; f1′N+1(t0) = 0, f1

′N+1(tN ) = 1.

(2.15)

Let g be the original function providing a data {(tj, xj)}N+1j=−1, where x−1 = g(t−1) =

g′(t0), xN+1 = g(tN+1) = g′(tN ). Let gc denotes the cubic spline CHFIF of Type-Ito this data. Let X (I,∆t, α, β, γ) = {h | h is a cubic spline CHFIF of Type-Ion ∆t}. Define the semi-inner product map on X (I,∆t, α, β, γ)×X (I,∆t, α, β, γ)

212 A. K. B. CHAND

as < φ,ψ >=N+1∑

j=−1

φ(tj)ψ(tj). So, the set {f1i | i = −1, 0, 1, . . . , N + 1} is or-

thonormal in this semi-inner product space, and hence it is linearly independent.If h ∈ X (I,∆t, α, β, γ) interpolates the same data {(tj , xj)}

N+1j=−1, then due to the

uniqueness of the fixed point of Read-Bajraktarevic operator, h(t) =N+1∑

i=−1

xif1i(t).

Since none of the f1i is a linear combination of other cardinal spline CHFIFs,

therefore {f1i}N+1i=−1 is a basis for X (I,∆t, α, β, γ). Define a coalescence cubic spline

operator G : C2(I) → X (I,∆t, α, β, γ) as G(g) = gc such that

(2.16) gc(Lj(t)) =

N+1∑

i=−1

g(ti)f1i(Lj(t)) =

N+1∑

i=−1

xif1i(Lj(t)), t ∈ I, j = 1, 2, . . . , N.

According to (2.14) and (2.15), we have gc(t0) =N+1∑

i=−1

xif1i(t0) =N+1∑

i=−1

xiδi,0 = x0

and gc(tj) =N+1∑

i=−1

xif1i(tj) =N+1∑

i=−1

xiδi,j = xj for j = 1, 2, . . . , N . Also, g′c(t0) =

N+1∑

i=−1

xif1′i(t0) = x−1 = g′(t0) and g

′c(tN ) =

N+1∑

i=−1

xif1′i(tN ) = xN+1 = g′(tN ). Simi-

larly, for a basis of cubic spline CHFIF of Type-II, we assume f1′′i (t0) = f1

′′i (tN ) = 0

for i = 0, 1, . . . , N, x−1 = g(t−1) = g′′(t0), xN+1 = g(tN+1) = g′′(tN ) and coales-cence cubic fractal splines f1−1 and f1N+1 satisfy

f1−1(tj) = 0, j = 0, 1, . . . , N ; f1′′−1(t0) = 1, f1

′′−1(tN ) = 0,

f1N+1(tj) = 0, j = 0, 1, . . . , N ; f1′′N+1(t0) = 0, f1

′′N+1(tN ) = 1.

(2.17)

Remark 2.1. (i) If αj = βj = 0, j = 1, 2, . . . , N, F 1i,j(t, x, ξ) reduces to a cubic

polynomial in t for the IFS (2.13) in each subintervals of I. Hence, in this case f1ireduces to a classical cardinal spline Si of Type-I (Type-II) such that Si(tj) = δi,j .

The classical complete cubic spline S(t) for the data {(tj , xj)}N+1j=−1 is given by

(2.18) S(Lj(t)) =N+1∑

i=−1

xiSi(Lj(t)), t ∈ I, j = 1, 2, . . . , N.

(ii) In general, f1i is not self-affine as it is the projection of the attractor of annon-diagonal IFS. But, if xi,j = ξi,j for i, j = 0, 1, 2, . . . , N and αj+βj = γj for j =1, 2, . . . , N, and f1i , f2i have the same boundary conditions, then the cardinal cubicspline CHFIF is self-affine in nature. Hence, it is possible to construct non-self-affine or self-affine bases for X (I,∆t, α, β, γ) with a suitable choice of parametersof the generalized IFS.

2.4. Optimal error bound associated with a cubic spline CHFIF. An up-per bound of L∞-norm of the error between a cubic spline CHFIF and the originalfunction g ∈ C4(I) is obtained with the help of the classical cubic spline, wherethe distance between the nodes is uniform. Denote α = (α1, α2, . . . , αN ), β =(β1, β2, . . . , βN ), γ = (γ1, γ2, . . . , γN ), |α|∞ = max{|αj| : j = 1, 2, . . . , N} and|β|∞ = max{|βj| : j = 1, 2, . . . , N}. We need the following proposition to get anupper bound of the error between the cardinal cubic spline and the correspondingcardinal spline CHFIF for the same data:

Proposition 2.3. Let f1i and Si (i = −1, 0, . . . , N + 1) be the cardinal cubicspline CHFIF and the classical cardinal cubic spline respectively to the same data

COALESCENCE CUBIC SPLINE FIS 213

{(ti, δi,j)}Ni=0 and same boundary conditions. Suppose f2i is the corresponding self-

affine cubic fractal function used in the construction of f1i. Let h = tj − tj−1, j =1, 2, . . . , N , a = h/|I|, and |I| = tN − t0. Suppose, there exist positive constantsK∗

r,i,K∗∗r,i, r = 0, 1, 2 such that

∣

∣

∣

∣

∂1+rpi,j(λj , βj , t)

∂αj∂tr

∣

∣

∣

∣

≤ K∗r,i,

∣

∣

∣

∣

∂1+rpi,j(αj , µj , t)

∂βj∂tr

∣

∣

∣

∣

≤ K∗∗r,i,

for |λj | ∈ (0, κ∗a2), |µj | ∈ (0, κa2), t ∈ Ij , r = 0, 1, 2 and j = 1, 2, . . . , N. Then forr = 0, 1, 2:

(2.19) ‖f1(r)i − S

(r)i ‖∞ ≤

h2−r[|α|∞(‖S(r)i ‖∞ +K∗

r,i) + |β|∞(‖f2(r)i ‖∞ +K∗∗

r,i)]

|I|2−r − h2−r|α|∞.

The proof of the above proposition can be seen in the reference [27]. From (2.16),(2.18) and (2.19), an upper bound for the error between the classical cubic splineof Type-I (Type-II) and corresponding cubic spline CHFIF of Type-I (Type-II) isgiven by for t ∈ I and r = 0, 1, 2:

|g(r)c (Lj(t)) − S(r)(Lj(t))| = |

N+1∑

i=−1

g(ti)(f1(r)i − S

(r)i )(Lj(t))|

≤

N+1∑

i=−1

‖g‖2‖f1(r)i − S

(r)i ‖∞,

≤

N+1∑

i=−1

‖g‖2h2−r[|α|∞(Θr +K∗

r ) + |β|∞(Γr,γ +K∗∗r )]

|I|2−r − h2−r|α|∞,

where C2-norm of g is ‖g‖2 = max{‖g‖∞, ‖g′‖∞, ‖g

′′‖∞}, Θr = max{‖S(r)i ‖∞ : i =

−1, 0, . . . , N + 1}, Γr,γ = max{‖f2(r)i ‖∞ : i = −1, 0, . . . , N + 1}, K∗

r = max{K∗r,i :

i = −1, 0, . . . , N + 1}, and K∗∗r = max{K∗∗

r,i : i = −1, 0, . . . , N + 1}. It can beverified that Θr is uniformly bounded as N → ∞ [28] and Γr,γ is uniformly bounded

as N → ∞ [27]. Denote Λr,α,β,γ,N =(N+3)[|α|∞(Θr+K∗

r)+|β|∞(Γr,γ+K∗∗

r)]

|I|2−r−h2−r|α|∞. Since the

above inequality is true for j = 1, 2, . . . , N , we have

(2.20) ‖g(r)c − S(r)‖∞ ≤ ‖g‖2Λr,α,β,γ,Nh2−r, r = 0, 1, 2.

The optimal error bound between the classical cubic spline S of Type-I (Type-II)and the original function g is known from the following proposition by Hall andMeyer [29]:

Proposition 2.4. Let S be the cubic spline interpolant of Type-I or Type-II to thegiven function g ∈ C4[t0, tN ] where tj − tj−1 = h for j = 1, 2, . . . , N. Then,

(2.21) ‖S(r) − g(r)‖∞ ≤ Cr‖g(4)‖∞h

4−r, r = 0, 1, 2

with C0 = 5/384, C1 = 1/24, C2 = 3/8.

From (2.20) and (2.21), we have the following optimal upper bound estimationfor the error if we approach the estimation through the classical cubic spline:

(2.22) ‖g(r)c − g(r)‖∞ ≤ ‖g‖2Λr,α,β,γ,Nh2−r + ‖g(4)‖∞Crh

4−r, r = 0, 1, 2.

If we define C4-norm ‖g‖4 =Max{‖g‖∞, ‖g′‖∞, . . . , ‖g

(4)‖∞}, then

(2.23) ‖g(r)c − g(r)‖∞ ≤ ‖g‖4[Λr,α,β,γ,N + Crh2]h2−r, r = 0, 1, 2.

214 A. K. B. CHAND

Remark 2.2. In (2.22), re-writing Λr,α,β,γ,N = 1hC(α, h)A(r, α, β, γ), where C(α, h)

= |I|+3h|I|2−r−h2−r |α|∞

and A(r, α, β, γ) = |α|∞(Θr+K∗r )+ |β|∞(Γr,γ +K

∗∗r ). Note that

as h→ 0, C(α, h) is bounded. The free scale vector α and the constrained scale vec-tor β are chosen for a given interpolation data, i.e., when h is known. Therefore,if we choose |α|∞ ≤ h and |β|∞ ≤ h in the construction of the IFS (2.13) for eachcardinal cubic spline CHFIFs, then the right hand side of (2.22) exists for r = 2and tends to zero for r = 1 whenever h → 0. Similarly, if there exists a positivereal µ such that |α|∞ ≤ h1+µ and |β|∞ ≤ h1+µ, then the cubic spline CHFIF ofType-I or Type-II gc converges to the original function g in C2-norm.

3. Coalescence Cubic Fractal Interpolation Surfaces

The construction of CCFISs has given over a rectangular grid D through thetensor product of univariate basis of coalescence fractal cubic splines of Type-Ior Type-II in Section 3.1. For an original function f ∈ C4[D], upper bounds ofL∞-norm of the errors between CCFIS and f , and their derivatives are deduced inSection 3.2.

3.1. Construction of CCFISs. Let f ∈ C2[D] be the original function, whereC2[D] is the family of real valued functions of two variables such that their 2nd orderpartial derivatives with respect to each single variable, exist and are continuous.Let ∆t : t0 < t1 < · · · < tN and ∆s : s0 < s1 < · · · < sM form a rectangularmesh Π : ∆t × ∆s for a rectangular region D = I × J , where J = [s0, sM ]. NowD is partitioned into a union of rectangles Di,j = [tj−1, tj ] × [sm−1, sm] ≡ Ij ×Jm, j = 1, 2, . . . , N ;m = 1, 2, . . . ,M . Suppose h = tj − tj−1, j = 1, 2, . . . , N andh = sm − sm−1, m = 1, 2, . . . ,M. Let Cf (t, s) be a CCFIS associated with thefunction f(t, s) and the mesh Π. Then, Cf (t, s) is a tensor product of cardinalcubic CHFIFs such that

(3.1)

Cf (tj , sm) = f(tj , sm); j = 0, 1, . . . , N ; m = 0, 1, . . . ,M,

C(k,0)f (tj , sm) = f (k,0)(tj , sm); j = 0, N ; m = 0, 1, . . . ,M,

C(0,k)f (tj , sm) = f (0,k)(tj , sm); j = 0, 1, . . . , N ; m = 0,M,

C(k,k)f (tn, sj) = f (k,k)(tn, sj); j = 0, N ; m = 0,M,

where C(p,q)f = ∂p+qCf/∂t

p∂sq and k = 1 (k = 2) if cardinal cubic CHFIFs are of

Type-I (Type-II). In this construction, two sets of nodal bases for univariate cubic

spline CHFIFs are needed. Let {f1i(t)}N+1i=−1 be a nodal basis for the cubic spline

CHFIF space X (I,∆t, α, β, γ) (cf. Section 2.3) and {f1n(s)}M+1n=−1 be a nodal basis

for the cubic spline CHFIF space X (J,∆s, α, β, γ) with IFS parameters: (i) αm,(ii) βm (iii) γm (iv) ξn,m (v) the boundary conditions of self-affine fractal functionf2n(s). Due to uniform partition of J , suppose that Lm : J → Jm takes the form

Lm(s) = as + bm for m = 1, 2, . . . ,M, where a = h|J| , and |J | = sM − s0. Define

generic transformations Φ and Ψ on C2[D] respectively as

(3.2) Φ(g(Lj(t), Lm(s))) =

N+1∑

i=−1

g(ti, Lm(s))f1i(Lj(t)),

(3.3) Ψ(h(Lj(t), Lm(s))) =

M+1∑

n=−1

h(Lj(t), sn)f1n(Lm(s)).

COALESCENCE CUBIC SPLINE FIS 215

The tensor product of the spaces of X (I,∆t, α, β, γ) and X (J,∆s, α, β, γ) with thebasis {f1i(t) f1n(s) | i = −1, 0, 1, . . . , N +1;n = −1, 0, 1, . . . ,M +1} is now definedas

X (I,∆t, α, β, γ)⊗X (J,∆s, α, β, γ) ={

N+1∑

i=−1

M+1∑

n=−1

ηi,nf1i(Lj(t))f1n(Lm(s)) | ηi,n ∈ R}

.

Finally, the CCFIS Cf = (Φ ◦Ψ)f for f ∈ C2[D] is defined as

(3.4) Cf (Lj(t), Lm(s)) =

N+1∑

i=−1

M+1∑

n=−1

f(ti, sn)f1i(Lj(t))f1n(Lm(s)), (t, s) ∈ D,

where f(t−1, s) = ∂f(t0, s)/∂t for Type-I or f(t−1, s) = ∂2f(t0, s)/∂t2 for Type-

II, with analogue notations for f(t, s−1), f(tN+1, s), f(t, sM+1), f(t−1, s−1), etc,depending on boundary conditions. It is easy to see that the function Cf satisfies theinterpolation conditions: From (3.4), for j ∈ {1, 2, . . . , N} and m ∈ {1, 2, . . . ,M},

Cf (tj , sm) = ((Φ ◦Ψ)f)(Lj(tN ), Lm(sM )) =

N+1∑

i=−1

M+1∑

n=−1

f(ti, sn)f1i(tj)f1n(sm)

=N+1∑

i=−1

M+1∑

n=−1

f(ti, sn)δi,jδn,m = f(tj , sm).

Similarly, Cf (t0, sm) = f(t0, sm), m = 1, 2, . . . ,M ; Cf (tj , s0) = f(tjs0), j =1, 2, . . . , N, and Cf (t0, s0) = f(t0, s0). Also, derivative conditions at the boundaryof CCFIS can be verified by using (2.15) or (2.17) depending on the type of cardinalcubic CHFIFs used in the construction. For instance if k = 2, for j = 0, N ;m =

0, 1, . . . ,M, C(2,0)f (tj , sm) =

N+1∑

i=−1

M+1∑

n=−1f(ti, sn)f1

′′i (tj)f1n(sm) = f (2,0)(tj , sm) as we

can obtain f1′′i (tj) = 0 for j = 0, 1, . . . , N from (2.17). Since f1i = Si if αj = βj =

0 ∀ j = 1, 2, . . . , N, and f1n = Sn if αm = βm = 0 ∀ m = 1, 2, . . . ,M, we canretrieve the classical cubic spline surface Sf to the original function f from (3.4).

3.2. Upper bounds of errors for CCFISs. We need the following notations toobtain the error bound of the original function g with the cubic spline CHFIF gcdefined on J . Suppose that pn,m; m = 1, 2, . . . ,M are cubic polynomials associatedwith the IFS for f1n such that

∣

∣

∣

∣

∂1+rpn,m(λm, βm, s)

∂αm∂sr

∣

∣

∣

∣

≤ K∗r,n,

∣

∣

∣

∣

∂1+rpn,m(αm, µm, s)

∂βm∂sr

∣

∣

∣

∣

≤ K∗∗r,n,

for |λm| ∈ (0, κ∗a2), |µm| ∈ (0, κa2), s ∈ Jm, r = 0, 1, 2, and m = 1, 2, . . . , N.

Denote Θr = max{‖S(r)n ‖∞ : n = −1, 0, . . . , M + 1}, Γr,γ = max{‖f2

(r)n ‖∞ : n =

−1, 0, . . . ,M+1}, K∗r = max{K∗

r,n : n = −1, 0, . . . ,M+1}, K∗∗r = max{K∗∗

r,n : n =

−1, 0, . . . ,M +1} and Λr,α,β,γM =(M+3)[|α|∞(Θr+K∗

r)+|β|∞(Γr,γ+K∗∗

r)]

|J|2−r−h2−r |α|∞, where f2n is

the associated cubic fractal function in the construction of f1n. Hence, similar to(2.22), we have the following estimation:

(3.5) ‖g(r)c − g(r)‖∞ ≤ [‖g‖2Λr,α,β,γ,M + ‖g(4)‖∞Crh2]h2−r, r = 0, 1, 2.

Denote ‖f‖(2,.) = max{‖f (0,.)‖∞, ‖f(1,.)‖∞, ‖f

(2,.)‖∞}, ‖f‖(.,2) = max{‖f (.,0)‖∞,

‖f (.,1)‖∞, ‖f(.,2)‖∞}, and ‖f‖(2,2) = max{‖f (p,q)‖∞ : p = 0, 1, 2; q = 0, 1, 2}.

216 A. K. B. CHAND

Theorem 3.1. Let Cf be the CCFIS to the original function f ∈ C4[D] := {g :

D → R : g(p,q) ∈ C[D], 0 ≤ p, q ≤ 4} defined on rectangular partitions of D,where h and h are fixed step lengths in t and s directions respectively. Then forp = 0, 1, 2; q = 0, 1, 2:

‖(f − Cf )(p,q)‖∞ ≤[‖f‖(2,q)Λp,α,β,γ,N + ‖f (4,q)‖∞Cph

2]h2−p + [‖f‖(p,2)Λq,α,β,γ,M

+ ‖f (p,4)‖∞Cqh2]h2−q +

[

(‖f‖(2,2) + ‖f‖(4,2)Cph2)Λq,α,β,γ,M

+ (‖f‖(2,4) + ‖f (4,4)‖∞Cph2)Cqh

2]

h2−ph2−q.

Proof. Consider

(3.6) f − Cf = (f −Ψ(f))− {f −Ψ(f)− (Φ(f)− Cf )}+ (f − Φ(f)).

Now,

(f − Φ(f))(Lj(t), Lm(s)) = f(Lj(t), Lm(s))−

N+1∑

i=−1

f(ti, Lm(s))f1i(Lj(t)).

For s = s∗ fixed, (Φ(f))(0,q)(., Lm(s∗)) is the cubic spline CHFIF of f (0,q)(., Lm(s∗)).Since f (0,q)(., Lm(s∗)) ∈ C4(I), therefore using (2.22), we have for p = 0, 1, 2,

‖(f − Φ(f))(p,q)(., Lm(s∗))‖∞ ≤ [max{‖f (0,q)(., Lm(s∗))‖∞, ‖f(1,q)(., Lm(s∗))‖∞,

‖f (2,q)(., Lm(s∗))‖∞}Λp,α,β,γ,N + ‖f (4,q)(., Lm(s∗))‖∞Cph2]h2−p.

Since the above inequality is true for all values of s∗ ∈ J , it gives that

(3.7) ‖(f − Φ(f))(p,q)‖∞ ≤ [‖f‖(2,q)Λp,α,β,γ,N + ‖f (4,q)‖∞Cph2]h2−p.

Similarly, using (3.5), we have for q = 0, 1, 2,

(3.8) ‖(f −Ψ(f))(p,q)‖∞ ≤ [‖f‖(p,2)Λq,α,β,γ,M + ‖f (p,4)‖∞Cqh2]h2−q.

Using definitions of Cf and Φ(f),

(Φ(f)− Cf )(Lj(t), Lm(s))

=N+1∑

i=−1

{

f(ti, Lm(s))−M+1∑

n=−1

f(ti, sn)f1n(Lm(s))}

f1i(Lj(t))

=

N+1∑

i=−1

U(ti, Lm(s))f1i(Lj(t)) = Φ(U(Lj(t), Lm(s))),

where U = f − Ψ(f). The middle term of (3.6) reduces to f − Ψ(f) − (Φ(f) −Cf ) = U − Φ(U). For s = s∗ fixed, Φ(U)(0,q)(., Lm(s∗)) is the cubic spline FIF

of U (0,q)(., Lm(s∗)). Using (2.22) for p = 0, 1, 2 (similar derivation as in (3.7)), weobtain

(3.9) ‖(U − Φ(U))(p,q)‖∞ ≤ [‖U‖(2,q)Λp,α,β,γ,N + ‖U (4,q)‖∞Cph2]h2−p.

For t = t∗ fixed,

U (k,q)(Lj(t∗), .) = f (k,q)(Lj(t

∗), .)−Ψ(f)(k,q)(Lj(t∗), .) for k = 0, 1, 2, 4.

Hence, for each k = 0, 1, 2, 4 by using (3.5) for q = 0, 1, 2, we have(3.10)

‖U (k,q)‖∞ ≤ [max{‖f (k,0)‖∞, ‖f(k,1)‖∞, ‖f

(k,2)‖∞}Λq,α,β,γ,M+‖f (k,4)‖∞Cqh2]h2−q.

COALESCENCE CUBIC SPLINE FIS 217

Hence, using (3.10) for k = 0, 1, 2,

‖U‖(2,q) = max{‖U (0,q)‖∞, ‖U(1,q)‖∞, ‖U

(2,q)‖∞}

≤ [‖f‖(2,2)Λq,α,β,γ,M + ‖f‖(2,4)Cqh2]h2−q,

(3.11)

and for k = 4,

(3.12) ‖U (4,q)‖∞ ≤ [‖f‖(4,2)Λq,α,β,γ,M + ‖f (4,4)‖∞Cqh2]h2−q.

Using (3.11) and (3.12) in (3.9), we have

‖(U − Φ(U))(p,q)‖∞ ≤ h2−p{

h2−q[‖f‖(2,2)Λq,α,β,γ,M + ‖f‖(2,4)Cqh2]

+ h2−q[‖f‖(4,2)Λq,α,β,γ,M + ‖f (4,4)‖∞Cqh2]Cph

2}

.(3.13)

The inequality of Theorem 3.1 follows from (3.6)- (3.8) and (3.13). �

Remark 3.1. (i) Now Λq,α,β,γ,M = 1hC(α, h)A(q, α, β, γ), where A(q, α, β, γ) =

|α|∞· (Θq + K∗q ) + |β|∞(Γq,γ + K∗∗

q ) (cf. Remark 2.2.) Hence, if we choose

max{|α|∞, |β|∞} ≤ h and max{|α|∞, |β|∞} ≤ h, then the upper bound of the errorof CCFIS exists in Theorem 3.1 for every h and h. This estimation is valid forboth self-affine and non-self-affine CCFISs.(ii) If β = 0 and β = 0, then the CCFIS is self-affine in nature and Theorem 3.1gives the convergence results for the self-affine FIS to the original function [18].In this case, if there exist positive reals µ and ν such that |α|∞ ≤ h1+µ and|α|∞ ≤ h1+ν , then the self-affine FIS converges to the original function in C2-norm as max{h, h} → 0.(iii) Suppose that there exist positive reals µ and ν such that max{|α|∞, |β|∞} ≤h1+µ and max{|α|∞, |β|∞} ≤ h1+ν , then the CCFIS Cf converges to the originalfunction f in C2-norm as max{h, h} → 0.(iv) When α = β = 0 and α = β = 0, then a CCFIS reduces to the classical splinesurface. In this case, Theorem 3.1 coincides with the error bound of the classicalspline surface [30].

4. Examples

We discuss examples of CCFISs based on cardinal cubic spline CHFIFs of Type-Ias the effects of the IFS parameters are very much similar for CCFISs based oncardinal cubic spline CHFIFs Type-II. We construct six sets of cardinal cubic fractalbases first with t0 = 1, t1 = 2, t2 = 3, I = [1, 3] and N = 2. For simplicity, wehave taken ξi,j = ξj for all i in the generalized data set. The cardinal interpolationdata are extended to R

3 with a choice of hidden variables : ξ1 = 10, ξ2 = 2, ξ3 = 15and boundary conditions f ′

2(t0) = −3, f ′2(t2) = 3. The standard cardinal cubic

fractal basis of Type-I (Set I) is constructed with αj = 0.8, βj = 0.35, γj = 0.4 forj = 1, 2. In our examples, L1(t) = 1

2 t +12 and L2(t) =

12 t +

32 . Polynomials p(t)

and q(t) are evaluated by using solutions of the system of equations described inSection 2.2. Using the IFS (2.13), the cardinal basis is generated and plotted inFig. 1(a) as per Section 2.3 . In the Set II, Set III and Set IV, we modify α, βand γ respectively with respect to the data in Set I. The effects of αj = −0.6 fromαj = 0.8, the effects of βj = 0.55 from βj = 0.35, and the effects of γj = −0.6from γj = 0.4 on the cardinal cubic CHFIFs basis can be observed from Fig. 1(b),Fig. 1(c), and Fig. 1(d) respectively. In the cardinal basis Set V, we modify onlyboundary conditions as f ′

2(t0) = 5, f ′2(t2) = −8 with respect to the data in Set

I and the corresponding cardinal cubic spline CHFIF basis is given in Fig. 1(e).Next, we modify the hidden variable ξ as ξ1 = −2, ξ2 = 9, ξ3 = 3 in the generalized

218 A. K. B. CHAND

-1.5

-1

-0.5

0

0.5

1

1 1.5 2 2.5 3

f1-1f10f11f12f13

(a) Set I : A standard cardinal fractalbasis.

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

1 1.5 2 2.5 3

f1-1f10f11f12f13

(b) Set II : Effect on basis by α.

-2

-1.5

-1

-0.5

0

0.5

1

1 1.5 2 2.5 3

f1-1f10f11f12f13

(c) Set III : Effect on basis by β.

-1

-0.5

0

0.5

1

1.5

2

1 1.5 2 2.5 3

f1-1f10f11f12f13

(d) Set IV : Effect on basis by γ.

-1.5

-1

-0.5

0

0.5

1

1.5

1 1.5 2 2.5 3

f1-1f10f11f12f13

(e) Set V : Effect on basis by boundaryconditions of f2.

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 1.5 2 2.5 3

f1-1f10f11f12f13

(f) Set VI : Effect on basis by ξ.

Figure 1. Bases for cardinal cubic CHFIFs of Type-I.

data with respect to Set I, and the corresponding cardinal cubic fractal basis isgenerated in Fig. 1(f). If α 6= 0, γ 6= 0 and β 6= 0, then f1i is called a cardinal

cubic spline CHFIF because f1′′

i is a typical CHFIF that may be self-affine or non-self-affine depending on the IFS parameters. These cardinal cubic CHFIFs differfrom their classical interpolants in the sense that they obey a functional relationand their fractal dimensions are non-integer in general. We know that the shapes ofthe classical cardinal cubic splines are fixed. But cardinal cubic CHFIFs providesinfinite number of shapes to its user by varying its parameters in the generalizedIFS. Importantly, if α = 0 and β = 0, we can retrieve the basis elements for theclassical cubic spline space. Using the above sets of cardinal cubic CHFIF bases,we have constructed various examples of CCFISs in the following.

COALESCENCE CUBIC SPLINE FIS 219

Table 1. Interpolation data for CCFISs.

f(tn, sj) s0 = 1 s1 = 2 s2 = 3t0 = 1 6 7 4t1 = 2 5 9 8t2 = 3 4 5 6

Assume that N = M = 2 and the interpolation data set is given in Table1. In all our examples, we assume the same boundary conditions for CCFISs asf (1,0)(tj , sm) = 5; j = 0, 2, m = 0, 1, 2, f (0,1)(tj , sm) = 3; j = 0, 1, 2, m =

0, 2, and f (1,1)(tj , sm) = 2; j,m = 0, 2. We use the tensor product of cardinalcubic CHFIFs bases in t and s-directions to generate CCFISs as described in (3.4)through iterations. By using cardinal bases Set I to Set VI, six different CCFISsare generated in Fig. 2 to Fig. 7 respectively, where the same basis is used inboth t and s-directions. Since the basis functions are similar in shapes in Set I,Set III and Set V, the effects of β and effects of boundary conditions on f2 are noteffective, and corresponding CCFISs in Fig. 2, Fig. 4 and Fig. 6 are very muchsimilar in nature. If α or β is deviated too much from the initial values, then we cannotice significant variations in the shape of a CCFIS. Hence, if |f1(tj)| > |f2(tj)| (|f2(tj)| > |f1(tj)| ) for all j = 0, 1, . . . , N , then α (β ) dominates its effects on aCCFIS in compare with β (α) respectively, else it is difficult to say the dominanceeffects of α or β on the shape of a CCFIS. The effects of boundary conditions on f2gives little variation on the overall shape of a CCFIS. It can be observed from Fig.5 that the shape of the CCFIS (in comparison with Fig. 2) is completely changedwith a large deviation of γ, and hence the hidden variable γ also play significantrole in the shape of a CCFIS. Fig. 7 illustrates that any significant change in thehidden variable ξ gives unpredictable shape of on a CCFIS, and this can be usefulin the simulation of smooth surfaces to get completely different shapes only byvarying the hidden variable ξ in the generalized IFS data.

Now we discuss some examples of CCFISs, where the bases in t and s directionsare different. First, we take Set III in t-direction, Set II in s-direction and thecorresponding CCFIS is generated in Fig. 8. The combine effects of α − β on aCCFIS can be observed from Fig. 8 in comparison with Fig. 2. Again, we takeSet IV in t-direction ; Set II and Set III in s-direction so that the combine effectsof α − γ and β − γ on the CCFIS are shown in Fig. 9 and Fig. 10 respectively.Since the effects of boundary conditions on f2 and the effects of β give similar typeof cardinal cubic fractal basis, the effects of the former hidden variable is similarto the effects of the latter. Next, we choose Set VI in t-direction ; Set III and SetIV in s-direction so that the combine effects of hidden variables ξ − β and ξ − γon the CCFIS are illustrated in Fig. 11 and Fig. 12. Also, if we take β = 0, thenthe corresponding CCFIS is self-affine in nature. Finally, we assume α = β = 0,then we retrieve the classical cubic spline surface in Fig. 13. Hence, from theseexamples it is clear that the free variable α, constrained free variable β, and hiddenvariables: γ, ξ and the boundary conditions of self-affine fractal function f2 playimportant role in the shape of a CCFIS. These variables are responsible for overallshape of fractal spline approximants and they provide an additional advantage forCCFISs over their classical counterparts as well as self-affine FISs. An infinitenumber of CCFISs can be constructed interpolation the same data by varyingthese hidden variables and it may be helpful in optimization problems, when weput additional conditions on the interpolation data. Since CCFISs are self-affine or

220 A. K. B. CHAND

Fig. 2 : CCFIS by using the basis asSet I in both directions.

Fig. 3 : CCFIS by using the basis asSet II in both directions.

Fig. 4 : CCFIS by using the basis asSet III in both directions.

Fig. 5 : CCFIS by using the basis asSet IV in both directions.

Fig. 6 : CCFIS by using the basis asSet V in both directions.

Fig. 7 : CCFIS by using the basis asSet VI in both directions.

non-self-affine, it can be useful in engineering applications like surface modelling,computer graphics, data visualization and CAGD.

COALESCENCE CUBIC SPLINE FIS 221

Fig. 8 : α− β effects on CCFIS. Fig. 9 : α− γ effects on CCFIS.

Fig. 10 : β − γ effects on CCFIS. Fig. 11 : ξ − β effects on CCFIS.

Fig. 12 : ξ − γ effects on CCFIS.Fig. 13 : Classical cubic spline

surface.

5. Conclusion

The construction of bases for Type-I and Type-II cubic spline CHFIFs is in-troduced in the present work. These cardinal cubic fractal splines are constructedthrough solutions of system of equations. Using tensor product of cardinal cubicCHFIFs, the coalescence cubic FIS is introduced over rectangular domains, where

222 A. K. B. CHAND

the partitions in t-direction is uniform, and s-direction is also uniform. These uni-form partitions are used to obtain upper of bounds of L∞-norm of the errors ofa CCFIS Cf with the original function f ∈ C4[D], and their derivatives. SinceCCFISs do not have explicit expressions, these error bounds can not be improvedif we estimate them through the classical cubic surface. In general, ∂4Cf/∂t

2∂s2 isa typical 2-dimensional fractal surface which is non-self-affine in nature, and hencewe say that the corresponding CCFIS is also non-self-affine. But if we choose β = 0and β = 0, then the CCFIS Cf is self-affine in nature. Also, if α = β = 0 andα = β = 0, then cardinal cubic CHFIFs are reduced to the classical cardinal cubicsplines so that the CCFIS Cf is reduced to the classical cubic surface, and thisshows the power of generalization in fractal methodology. The C2-convergence re-sults of the CCFIS to the original function is obtained through proper restrictionson the scale vector and the constrained scale vector. The effects of free variables,constrained free variables and hidden variables are demonstarted through variousexamples. The presence of hidden variables can be utilised in bivariate optimizationproblems with given interpolation conditions. The present work may play impor-tant roles in smooth surface modelling in computer graphics, engineering design,and other smooth surface approximation problems in science and engineering.

Acknowledgements

The work is supported by the project No: MAT/08-09/234/NFSC/ARYA, IITMadras, India. The author is thankful to the anonymous reviewer for valuablesuggestions on the convergence results.

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Department of Mathematics, Indian Institute of Technology Madras Chennai 600036, India.E-mail : chand@iitm.ac.inURL: http://mat.iitm.ac.in/home/chand/public html/index.html